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OBJECTIVE: To see if the result of applying three forces on an object can be determined by ADDING the three forces as VECTORS.
GENERAL PROCEDURE: In this experiment your lab instructor will assign you one set of three vectors from each of the two groups listed on the fourth page. For each of these two vector sets, add the vectors to get the result using all three methods below (Parts 1,2,3). Be sure to estimate your uncertainty in any experimental measurement. Part 4 will lead you through a method of determining the overall uncertainty of your results. The last two pages will show you an example of a written report for this experiment.
Part 1: Analytical Method
THEORY: To add two or more vectors together, it is first necessary to express each of the vectors in rectangular components, ( Ax,Ay ). If a vector is expressed in polar form ( A ,), where A is the magnitude and is the angle indicating direction, then the rectangular components can be found using the basic trigonometry of a right triangle:
Ax A cos Ay A sin (1)
Then all the x-components of the vectors are added together to obtain the resultant x-component, Rx , all the y-components are added to obtain the resultant y-component, Ry , and if in 3-D, the same goes for the z-components:
Rx Ax R y Ay (2)
Once the rectangular components of the resultant vector have been obtained, the magnitude R of the resultant vector can be obtained using the Pythagorean Theorem:
R R (^) x^2^ Ry^2 (3)
The angle that the resultant vector makes with the x-axis, R , can be determined using the basic trigonometry of a right triangle:
tan 1 ^ Ry Rx (4)
Be careful: If Rx is negative, q is in the 2nd^ or 3rd^ quadrant. If the tan-1^ function returns an angle in the 1st^ or 4th quadrant when Rx is negative, you will need to add 180o^ to the angle so that it is in the correct quadrant.
PROCEDURE: Add each of the two sets of three vectors analytically. Express the resultant vectors in polar form, i.e. find the magnitude and angle of each of the resultant vectors.
Part 2: Graphical Method
THEORY: To add two or more vectors together graphically, it is first necessary to set up a coordinate system. For vectors expressed in polar coordinates, an origin and a horizontal line are needed. Then each vector can be expressed as a directed line segment on this graph. The length of the line is determined by the magnitude of the vector according to some scale. The direction of the line is determined by having the line segment directed at an angle measured counterclockwise from the horizontal coordinate line (or from some line parallel to the horizontal coordinate line) where is equal to the angle of the vector. The first vector starts at the origin. The next vector starts at the tip of the first vector. And each additional vector to be added starts at the tip of the preceding vector. The resultant is then obtained by drawing in a line segment from the origin (tail of the first vector) to the tip of the last vector. The length of this segment then gives the magnitude of the resultant vector, and the angle measured from the horizontal axis gives the angle of the resultant vector.
PROCEDURE: Add each of the two sets of three vectors graphically. Warning: placing the origin in the middle of the paper is not always the best - you may run out of room on the paper! Use a scale of 1 cm for 20 grams. Estimate how accurately you can draw and measure each length and angle.
Part 3: Experimental Method (Using the Force Table)
THEORY: The three forces representing the three vectors to be added are applied to the ring on the force table by setting a pulley on the force table at the angle corresponding to each of the three vectors and then attaching weights corresponding to the magnitude of each vector to strings hung over the pulleys. The angle and weight needed to balance the system and keep the ring centered on the post determine a fourth vector, E , such that all four vectors add to the zero vector: A + B + C + E = 0 (5) or, A + B + C = - E = R (6)
Therefore the resultant vector, R , is equal in magnitude to E but oppositely directed. Thus the angle determined on the force table is 180 different from the angle of the resultant vector.
PROCEDURE: Add each of the two sets of three vectors using the force table. (NOTE: The weight holder has a weight of 50 grams. Should this be included in your weight determination?) Estimate how well you can set and measure the angles, and estimate how well you can determine the mass of the fourth vector E.
qA = 0
A (^) q B
B
C q C
R
qR
VECTOR TABLE
A B C
SET (Mag,Angle) (Mag,Angle) (Mag,Angle) Group I 1 (200 gm, 0°) (100 gm, 70°) (100 gm, 135°) 2 (200 gm, 0°) (150 gm, 80°) (100 gm, 140°) 3 (150 gm, 0°) (100 gm, 45°) (150 gm, 120°) 4 (100 gm, 0°) (200 gm, 80°) (150 gm, 135°) 5 (150 gm, 0°) (200 gm, 60°) (150 gm, 150°)
Group II 6 (200 gm, 0°) (150 gm, 300°) (100 gm, 190°)
7 (200 gm, 0°) (100 gm, 135°) (150 gm, 280°) 8 (150 gm, 0°) (200 gm, 140°) (150 gm, 200°) 9 (100 gm, 0°) (150 gm, 330°) (200 gm, 170°) 10 (100 gm, 0°) (200 gm, 200°) (150 gm, 135°)
(see the next two pages for an example of a formal written report for this lab)
Sample of a Written Lab Report
(You should use this as a guide when writing up your formal written lab reports.)
Object: To see if the result of applying three forces on an object can be determined by ADDING the three forces as VECTORS. ( Note that the original objective was used here.)
Data: Our group was assigned problems 5 and 6. For these problems, the three vectors were: Problem 5: A = (150 gm, 0o), B = (200 gm, 60o), and C = (150 gm, 150o) Problem 6: A = (200 gm, 0o), b = (150 gm, 300o), and C = (100 gm, 190o).
The first method, the analytical method, employed only calculations. No data were taken.
The second method, the graphical method, had us create graphs. See the attached graphs that we made. (No graphs were made for this example.)
The third method, the force table method, gave the following results:
Problem Vector A Vector B Vector C Vector E (the equilibrant)
5 150 gm at 0o^ 200 gm at 60o^ 150 gm at 150o^ 280g ± 10g at 245o^ ± 1o 6 200 gm at 0o^ 150 gm at 300o^ 100 gm at 190o^ 230g ± 10g at 141o^ ± 2o
(Note that a table was used for the data. This is highly recommended!)
Graphs: See the attached graphs showing the graphical method of adding vectors. (These graphs are not included in this sample report. Usually, data is recorded first and then put in graphs to visually show the relationships. Spreadsheets such as Excel provide a powerful and easy way of graphing data. Be sure to show all data points, and also include a best fit line. Include the equation of the best fit line, but also convert it from the y(x) format into a physics format that includes units for all values.)
Calculations: (Only one sample calculation for each type should be shown. Units should always be included in all calculations.) For the first, analytical, part, we have to first convert all the vectors from the given polar form into rectangular form:
Vector A: 150 gm at 0o^ : RA = 150 gms; θA = 0o AX = RA * cos(θA) = 150 gm * cos(0o) = 150 gm; Bx = 100.0 gm; Cx = -129.9 gm AY = RA * sin(θA) = 150 gm * sin(0o) = 0 gm. By = 173.2 gm; Cy = 75.0 gm
Next we have to add all of the x components to get the total x component: XR = Ax + Bx + Cx = 150.0 gms + 100.0 gms - 129.9 gms = 120.1 gms YR = Ay + By + Cy = 0.0 gms + 173.2 gms + 75.0 gms = 248.2 gms
